In the univariate (\(d = 1\) ) case,
(REF ) reduces to \(V_{\mathrm {mars}}(f) = \int _0^1 |f^{\prime \prime }|\) , and thus
\(\hat{f}_{n, V}^1\)  can be thought of as a
constrained analogue of the
locally adaptive regression splines estimator of
[1]} when the order \(k\)  (in the notation of
[1]}) equals 2. Thus, for \(d \ge 2\) ,
\(\hat{f}_{n, V}^d\)  can be seen as a multivariate generalization
of this univariate estimator of [1]}. Further, if the condition
\(\max _j \beta _j = 2\)  in (REF ) is replaced with \(\max _j\beta _j = 1\) , then one obtains the Hardy-Krause variation of \(f\) . As a result, we
can refer to \(V_{\mathrm {mars}}(f)\)  as a
second-order Hardy-Krause variation
anchored at \((0, \dots , 0)\) and the estimation procedure leading
to \(\hat{f}_{n, V}^d\)  as second-order Hardy-Krause
variation denoising. For comparison, a first-order Hardy-Krause
variation denoising estimator was studied in [4]}.
